On certain properties of branching coefficients for affine Lie algebras

Abstract:

It is demonstrated that the decompositions of integrable highest weight modules of a simple Lie algebra (classical or affine) with respect to its reductive subalgebra obey a set of algebraic relations leading to recursive properties for the corresponding branching coefficients. These properties are encoded in a special element $\Gamma _{\mathfrak {g} \supset \mathfrak {a}}$ of the formal algebra $\mathcal {E}_{\mathfrak {a}}$ that describes the injections $\mathfrak {a}\to \mathfrak {g}$ and is called a fan. In the simplest case where $\mathfrak {a} = \mathfrak {h} (\mathfrak {g})$, the recursion procedure generates the weight diagram of a module $L_{\mathfrak {g}}$. When the recursion described by a fan is applied to highest weight modules, it provides a highly efficient tool for explicit calculations of branching coefficients.